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You can learn Hausdorff measure and dimension in Fractal Geometry: Mathematical Foundations and Applications by Kenneth Falconer. There are beginner examples of computing fractal dimensions of sets as well as ones related to approximation of irrationals.

It occurred to me that you might find some very detailed and explicit Hausdorff dimension calculations in a Masters (or undergraduate Honors) Thesis, so I googled each of the phrases "Masters Thesis" and "Honors Thesis" with the phrase "Hausdorff dimension" and found the following: